A Modern Approach to Quantum Mechanics · Problem 9.17

Bound States of Central Potentials — John S. Townsend

Physics — Quantum Mechanics · spherical-harmonics-legendre-equation

Problem

Determine the spherical harmonics and the eigenvalues of L^2\hat{L}^2 by solving the eigenvalue equation L^2∣λ,m⟩=λℏ2∣λ,m⟩\hat{L}^2|\lambda, m\rangle = \lambda\hbar^2|\lambda, m\rangle in position space, −ℏ2[1sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1sin⁡2θ∂2∂ϕ2]Θλ,m(θ)eimϕ=λℏ2Θλ,m(θ)eimϕ-\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right]\Theta_{\lambda,m}(\theta)e^{im\phi} = \lambda\hbar^2\Theta_{\lambda,m}(\theta)e^{im\phi} Note that we have inserted the known ϕ\phi dependence. To illustrate the procedure, restrict your attention to the m=0m = 0 case. Rewrite the equation in terms of u=cos⁡θu = \cos\theta and show that it becomes (1−u2)d2Θλ,0du2−2udΘλ,0du+λΘλ,0=0(1-u^2)\frac{d^2\Theta_{\lambda,0}}{du^2} - 2u\frac{d\Theta_{\lambda,0}}{du} + \lambda\Theta_{\lambda,0} = 0 which you may recognize as Legendre's equation. Try a power series solution Θλ,0=∑k=0∞akuk\Theta_{\lambda,0} = \sum_{k=0}^{\infty} a_k u^k Show that the series diverges for ∣u∣→1|u| \to 1 (θ→0\theta \to 0 or θ→π\theta \to \pi) unless λ=l(l+1)\lambda = l(l+1) with l=0,1,2,…l = 0, 1, 2, \ldots, in which case the series terminates. The solutions Θl,0(u)=Pl(u)\Theta_{l,0}(u) = P_l(u) are just the Legendre polynomials. Compare the first few solutions with the spherical harmonics listed in equations (9.151), (9.152b), and (9.153c).

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